document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209655>Question 1209655</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The graph of the equation<BR>\n" );
document.write( "4x^2 - 12x + 4y^2 + 16y - 15 = 10x + 25y + 28<BR>\n" );
document.write( "is a circle.  Find the radius of the circle.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849748 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849748\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the radius of the circle:\r<BR>\n" );
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document.write( "1. **Rewrite the equation by grouping x and y terms:**\r<BR>\n" );
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document.write( "4x² - 12x + 4y² + 16y - 15 = 10x + 25y + 28<BR>\n" );
document.write( "4x² - 22x + 4y² - 9y = 43\r<BR>\n" );
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document.write( "2. **Complete the square for x and y:**\r<BR>\n" );
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document.write( "* **For x:**<BR>\n" );
document.write( "  4(x² - (11/2)x)  <BR>\n" );
document.write( "  To complete the square, take half of -11/2 which is -11/4 and square it to get 121/16.<BR>\n" );
document.write( "  4(x² - (11/2)x + 121/16) - 4(121/16)\r<BR>\n" );
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document.write( "* **For y:**<BR>\n" );
document.write( "  4(y² - (9/4)y)<BR>\n" );
document.write( "  To complete the square, take half of -9/4 which is -9/8 and square it to get 81/64<BR>\n" );
document.write( "  4(y² - (9/4)y + 81/64) - 4(81/64)\r<BR>\n" );
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document.write( "3. **Substitute the completed squares back into the equation:**\r<BR>\n" );
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document.write( "4(x² - (11/2)x + 121/16) - 4(121/16) + 4(y² - (9/4)y + 81/64) - 4(81/64) = 43<BR>\n" );
document.write( "4(x - 11/4)² - 121/4 + 4(y - 9/8)² - 81/16 = 43\r<BR>\n" );
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document.write( "4. **Simplify and rewrite in standard circle form:**\r<BR>\n" );
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document.write( "Multiply the entire equation by 16 to eliminate fractions:\r<BR>\n" );
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document.write( "64(x - 11/4)² - 484 + 64(y - 9/8)² - 81 = 688<BR>\n" );
document.write( "64(x - 11/4)² + 64(y - 9/8)² = 688 + 484 + 81<BR>\n" );
document.write( "64(x - 11/4)² + 64(y - 9/8)² = 1253\r<BR>\n" );
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document.write( "Divide by 64:<BR>\n" );
document.write( "(x - 11/4)² + (y - 9/8)² = 1253/64\r<BR>\n" );
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document.write( "5. **Identify the radius:**\r<BR>\n" );
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document.write( "The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.\r<BR>\n" );
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document.write( "In our equation, r² = 1253/64. Therefore, the radius is:\r<BR>\n" );
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document.write( "r = √(1253/64) = √1253 / 8\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{\frac{\sqrt{1253}}{8}}$<BR>\n" );
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